1. **State the problem:** Given the universal set size $n(U) = 144$, the size of set $A$ is $n(A) = 46$, the size of set $B$ is $n(B) = 83$, and the size of the union $n(A \cup B) = 129$, find the size of the intersection $n(A \cap B)$. 2. **Formula used:** The cardinality of the union of two sets is given by the formula: $$n(A \cup B) = n(A) + n(B) - n(A \cap B)$$ This formula accounts for the fact that elements in the intersection are counted twice when adding $n(A)$ and $n(B)$, so we subtract $n(A \cap B)$ once. 3. **Substitute the known values:** $$129 = 46 + 83 - n(A \cap B)$$ 4. **Simplify the right side:** $$129 = 129 - n(A \cap B)$$ 5. **Isolate $n(A \cap B)$:** $$129 - 129 = - n(A \cap B)$$ $$0 = - n(A \cap B)$$ 6. **Multiply both sides by $-1$ to solve for $n(A \cap B)$:** $$\cancel{0} = - \cancel{n(A \cap B)}$$ $$n(A \cap B) = 0$$ **Final answer:** $$\boxed{0}$$ This means sets $A$ and $B$ have no elements in common.